Q56P - University Physics with Modern Physics

Question

Biological tissues are typically made up of $98\%$ water. Given that the density of water is $1.0\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3$, estimate the mass of (a) the heart of an adult human; (b) a cell with a diameter of $0.5\mu \mathrm{m}$; (c) a honeybee.

Step-by-Step Solution

Answer

a) Estimated mass of a human heart is $128.5 \mathrm{g}$.

b) Estimated mass of a cell is $6.41\times {10}^{-14} \mathrm{g}$.

c) Estimated mass of the honeybee is $2.77 \mathrm{g}$.

Identification of given data

The given data is listed below,

• Percentage of water content in biological tissues is $98\%$.
• The density of water is, ${\rho }_w=1.0\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3$.
• The diameter of the cell is, $D_c=0.5\mu \mathrm{m}\left(\frac{{10}^{-6} \mathrm{m}}{1\mu \mathrm{m}}\right)=0.5\times {10}^{-6} \mathrm{m}$.

Concept of the mass and density

The Objects have mass as they are made up of atoms, and atoms have mass as they are made up of fundamental particles- Protons, Electrons and Neutrons. Mass is a fundamental property of matter due to which a body undergoes gravitational interactions.

(a) Determination of the mass of the heart of an adult human

The size of the human heart can be estimated roughly as the size of a fist.

Assume its volume is $(2\text{ in }{)}^3=8 {\mathrm{in}}^3$ 

Mass of human heart is given by,

$m=\rho V$

Here, $\rho$ is the density of water and $V$ is the volume of the heart.

Substitute values in the above expression and using the information that $98\%$ of the biological tissue is made up of water,

$\begin{array}{rcl}m& =& (0.98)\left(1.0\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3\right)\left(8{\mathrm{in}}^3\right)\\ & =& (0.98)\left(8\times {10}^3{\mathrm{in}}^3 \mathrm{kg}/{\mathrm{m}}^3\right){\left(\frac{2.54 \mathrm{cm}}{1\mathrm{in}}\right)}^3{\left(\frac{1 \mathrm{m}}{100 \mathrm{cm}}\right)}^3\\ & =& 0.12847 \mathrm{kg}\left(\frac{1000 \mathrm{g}}{1 \mathrm{kg}}\right)\\ & =& 128.5 \mathrm{g}\end{array}$

Thus, the mass of a human heart is estimated as $128.5 \mathrm{g}$.

(b) Determination of the mass of a cell

Assume the cell as a sphere and therefore the volume of the cell is given by,

$V=\frac{4\pi }{3}{r}^3$

Here, $r$ is the radius of the sphere.

Substitute values in the above,

$$\begin{gathered} V_c=\frac{4\pi }{3}{\left(\frac{0.5\times {10}^{-6} \mathrm{m}}{2}\right)}^3 \\ =6.54\times {10}^{-20}{ \mathrm{m}}^3 \end{gathered}$$

The mass of the cell is given by,

$m=\rho V$

Here, $\rho$ is the density of water and $V$ is the volume of the cell.

Substitute values in the above and using the information that $98\%$ of the biological tissue is made up of water,

$\begin{array}{rcl}{m}_c& =& (0.98)\left(1.0\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3\right)\left(6.54\times {10}^{-20}{ \mathrm{m}}^3\right)\\ & =& 6.41\times {10}^{-17} \mathrm{kg}\left(\frac{1000 \mathrm{g}}{1 \mathrm{kg}}\right)\\ & =& 6.41\times {10}^{-14} \mathrm{g}\end{array}$

Thus, the mass of a cell is estimated as $6.41\times {10}^{-14} \mathrm{g}$.

(c) Determination of the mass of a honeybee

Let the honeybee as a cylinder. and assume the length of the cylinder is $1 cm$ and the radius is $3 \mathrm{mm}$.

The volume of a cylinder is given by,

$V=\pi r^2/$

Here, $r$ is the radius of the cylinder whose value is assumed as $1 \mathrm{cm}$ and $l$ is the length of the cylinder whose value is assumed as $3 \mathrm{mm}$

Substitute values in the above,

$\begin{array}{rcl}V& =& \pi (3 \mathrm{cm}{)}^2(1 \mathrm{mm})\\ & =& \pi \times 9{ \mathrm{cm}}^2\times \mathrm{mm}\left(\frac{1 \mathrm{m}}{1000 \mathrm{mm}}\right){\left(\frac{1 \mathrm{m}}{100 \mathrm{cm}}\right)}^2\\ & =& 2.83\times {10}^{-6}{ \mathrm{m}}^3\end{array}$

The mass of the honeybee is given using the information that $98\%$ of the biological tissue is made up of water,

$\begin{array}{rcl}m& =& (0.98)\left(1\times {10}^3 \mathrm{kg}/{\mathrm{m}}^3\right)\left(2.83\times {10}^{-6}{ \mathrm{m}}^3\right)\\ & =& 2.77\times {10}^{-3} \mathrm{kg}\left(\frac{1000 \mathrm{g}}{1 \mathrm{kg}}\right)\\ & =& 2.77 \mathrm{g}\end{array}$

Thus, the mass of the honeybee will be $2.77 \mathrm{g}$.